Let $X$ be a CW complex then we may consider the simplicial set $EX_{\bullet} : [n] \mapsto X^{\times n+1}$ with the face maps being projections and degeneracies being diagonals. I'm pretty sure that the $n$-th geometric realization is homeomorphic to the $n$-fold join of $X$ with itself i.e. $|Sk^n (EX_{\bullet})| \cong X^{\ast n}$

Does anyone know a reference for this?

*Correction: its supposed to be $|Sk^n EX_{\bullet}| \cong X^{\ast n+1}$